I've generally found the "family suites" to not be a good deal...it seems that the 1st and 2nd fares are way too high, and those cabins (only a few per ship) are booked long in advance.
We had connecting doors once, but it seems there's very few of those cabins available either.
Generally we get a "boys cabin" for me and my son, and a "girls cabin" for my wife and daughters, right next to each other (or in some cases across from each other). We generally trade things around once aboard, so that the kids are in one cabin and us in the other.
It helps in this case to get an extra key to the kids room from the front desk. I hear others take the little "baby monitor" transmitter/receiver pair to let them monitor the kids room from next door, but as yours are 10, 16, and 19 we don't bother with that any more.